Foundations — Normal shock properties — M₂, P₂ - P₁, T₂ - T₁, ρ₂ - ρ₁, P₀₂ - P₀₁

Before you can trust equations like $M_2^2=\frac{1+\frac{\gamma -1}{2}{M}_1^2}{\gamma M_1^2-\frac{\gamma -1}{2}}$, you must know what each letter is, what picture it draws, and why the topic can't live without it. We go one symbol at a time, and never use a letter before it is earned.

1. Two states: the subscripts "1" and "2"

The whole topic is a before/after story. The gas comes in on one side of the shock and leaves on the other.

Picture it as a doorway. Everything to the left of the doorway is "state 1"; everything to the right is "state 2". Nothing about a shock makes sense until you've fixed which side you're standing on.

Why the topic needs it: every single equation in the parent note is a ratio like $P_2/P_1$ — a comparison of after to before. Without the labels, the ratios are meaningless.

2. Pressure $P$ — the push of the gas

Picture it as countless tiny molecules drumming against a wall. More drumming per second, or harder hits, means higher $P$. The teal arrows below show gas molecules bombarding a surface.

3. Density $\rho$ (rho) — how crowded the gas is

The symbol $\rho$ is the Greek letter "rho" (say "row"). It is not the letter p — that trips everyone up once.

Picture it as dots in a box. Few dots = low density (thin air); many dots crammed together = high density (thick air). Crossing a shock the dots get shoved closer, so ${\rho }_2>{\rho }_1$.

Why the topic needs it: density is what "compression" means. The famous result that ${\rho }_2/{\rho }_1$ can never exceed $6$ for air (equation 9's limit) is a statement about how tightly dots can be packed.

4. Temperature $T$ — the jiggle energy

Picture it as blurry, fast-vibrating dots (hot) versus slow, calm dots (cold). Squeeze and slow a supersonic flow and its ordered forward motion converts into random jiggle — so $T_2>T_1$. That is why hypersonic vehicles burn: the parent's Example 2 heats air from $300$ K to $1740$ K.

5. The ideal gas law — the glue between $P$, $\rho$, $T$

These three properties are not independent. They are locked together by one relation.

Picture it as a see-saw: hold volume fixed and heating the gas (raise $T$) makes it push harder (raise $P$); pack more dots in (raise $\rho$) and it also pushes harder. This is exactly how the parent gets temperature from the other two: $\frac{T_2}{T_1}=\frac{P_2/P_1}{{\rho }_2/{\rho }_1}$ (rearranged from $P=\rho RT$).

Why the topic needs it: it is the "equation of state" that closes the system — the parent says five unknowns need the three conservation laws plus $P=\rho RT$. This is that plus.

6. Velocity $u$ — how fast the gas travels forward

Picture it as a river's current: $u$ is the arrow showing the water's travel direction and speed. Across a shock the flow slows down, so $u_2<u_1$. Because mass is conserved, slowing the flow means crowding the dots — this is why ${\rho }_2{u}_2={\rho }_1{u}_1$ ties velocity and density together (parent equation 1).

7. Speed of sound $a$ — how fast news travels

Here is the pivot of the entire chapter. Sound is how a gas passes messages: "make room, something's coming."

Picture it as ripples spreading from a pebble dropped in a pond. If you move slower than the ripples, they race ahead and warn the water in front of you. If you move faster than the ripples, you outrun your own warning — the water ahead never sees you coming until you crash into it. That crash is the shock. The plum figure below shows both cases.

Why the topic needs it: the shock exists because the flow outruns its own sound. Without $a$, there is no way to say "faster than sound", and the whole idea collapses.

8. Mach number $M$ — speed measured in "sounds"

Picture it as a ratio dial: $M$ tells you how many "sound-speeds" fast you are going. The parent's master result (equation 7) is a machine that eats $M_1>1$ and always spits out $M_2<1$: supersonic in, subsonic out.

9. The ratio of specific heats $\gamma$ (gamma)

The symbol $\gamma$ is the Greek "gamma". For air it is $1.4$.

Picture it as the gas's "stiffness personality": it decides how readily pushing (pressure) turns into heating. You don't need to derive it here — just know it is a constant you carry along, appearing in $a=\sqrt{\gamma RT}$ and in the magic number $\frac{\gamma +1}{\gamma -1}=6$ that caps compression.

10. Stagnation properties $T_0$ and $P_0$ — the "brought to rest" values

Imagine gently, smoothly bringing the moving gas to a complete stop. All its forward motion becomes heat and pressure. The values it would then have are the stagnation (or "total") values, marked with subscript $0$.

Picture it as pressing your palm flat against the wind: the air right at your palm has stopped, and that stopped air is hotter and higher-pressure than the freely streaming air. The plum-and-orange figure shows a flow being smoothly decelerated to its stagnation point.

11. Entropy $s$ — the messiness meter

Picture it as a shuffled deck: you can shuffle order into chaos for free, but you can never un-shuffle without effort. A shock shuffles the neat supersonic flow, so $\Delta s>0$ always. That single fact forbids "backward" shocks and forces $P_{02}/P_{01}<1$. See Entropy and Second Law.

Prerequisite map

Every box on the left is a symbol built on this page; the arrows show how they feed into the parent topic on the right.

Equipment checklist

Cover the right-hand side and test yourself. If you can answer each, you are ready for the parent note.

Continue to the parent: Normal Shock Properties. Related building blocks: Isentropic Flow Relations, Rankine–Hugoniot Relations.